Sun Radiation Spectrum

Discounting the effect of Earth atmosphere and random solar flares, is it identical always? Do we get any variability in wavelengths due to elliptical nature of orbit or because of any other unexplained reason?

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Discounting the effect of Earth atmosphere and random solar flares, is it identical always?

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The spectrum remains similar. There is a moderate amount of variability (20% or so) in shortwave ultraviolet (below 200nm) due primarily to plages, which are the opposite of sunspots (i.e. brighter areas on the Sun.)

There are also very slight changes in:

-intensity, due to the sunspot cycle and our slightly elliptical orbit
-center frequency, due to the slight Doppler shift caused by our elliptical orbit

I read somewhere (not able to trace) that there is around 7% shift in the wavelengths due to orbital motion and most of that is unexplained. The radial speed doppler effect cannot account for that, so what else could be the cause for such high shift, if at all it is there.

2. Sun is losing energy daily. So as it ages it cools, and therefore it's radiation is red-shifted. This is from the "emitter's" .

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Wrong. Stars aren't as simple as that tacitly assumes. Our sun was considerably cooler in the past. Long term net output and mean temperature has been steadily increasing.
And that trend is predicted to continue for billions of years more: https://en.wikipedia.org/wiki/Sun#Life_phases
There are shorter term fluctuations as described there, the most well known being that linked to 11-year sunspot cycles.

This is not relevant. It is radiating energy, because it is continuing to consume fuel. The Sun will not run out of fuel for several billion years.

While that may be true, that doesn't exactly apply.

It's like saying "humans lose their hair as they age". Technically true, in the grand scheme, but it's hardly applicable to a teenager, is it?

Main sequence stars go through a very long period of stable temperatures before they begin to show their age.

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Our sun has been slowly increasing its output since it formed, and will continue to do show between now and when it begins to swell into a red giant. This is a natural process of its evolution. This a slow, slow increase ( 1 % increase per 100 million years or so) but pretty steady over the long run.
Here's a chart showing the the change in the Sun's temp, luminosity and radius over time.

It takes around 8 minutes for Sun light to reach Earth. During this period due to rotation of the Earth around the Sun, the relative position of the Earth has changed. What is the path traced by a photon which reaches earth? For example a photon emitted at any front side point of the surface of the sun, may skip the earth, even though the very same photon may hit the earth if there was no rotation.

My thinking (obviously wrong) was that, given an object of finite mass that is radiating (losing) energy, then its energy/mass content must be reduced over time - this is \(e=mc^2\).

I then used \(e=h\nu\) (\(\nu\) is frequency) and assumed that the lower energy content implies a reduced frequency.

Obviously there is something about fusion reactions I don't understand - hardly a surprise.......

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The basic concept goes likes this:
Hydrogen is fused into Helium at the core, which increases the percentage of Helium, which in turn in creases the density of the core, which in turn increases the pressure of the core and the rate of fusion. This generates more energy, which causes the Sun to expand. The Surface temp dos not change much, but because of the increased surface area, the luminosity of the Sun increases. This continues until the helium build up in the core actually starts to clog up the works, and you enter a shell burning stage where the hydrogen fusion occurs in a shell surrounding the Helium core. This marks the end of the main sequence part of its life.

It takes around 8 minutes for Sun light to reach Earth. During this period due to rotation of the Earth around the Sun, the relative position of the Earth has changed. What is the path traced by a photon which reaches earth? For example a photon emitted at any front side point of the surface of the sun, may skip the earth, even though the very same photon may hit the earth if there was no rotation.

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http://henry.pha.jhu.edu/aberration.html
There, that was easy enough. However, note that formula doesn't take into account rotation of the Earth. It can be factored in by applying the same principle though.

It takes around 8 minutes for Sun light to reach Earth. During this period due to rotation of the Earth around the Sun, the relative position of the Earth has changed. What is the path traced by a photon which reaches earth?

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A straight line.

For example a photon emitted at any front side point of the surface of the sun, may skip the earth, even though the very same photon may hit the earth if there was no rotation.

A photon emitted during the nuclear fusion process in the core of the Sun, takes thousands of years to reach the surface, ...

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Just to clarify on that popular but incorrect picture. In the QED interpretation, photons are continually being created and annihilated whenever they interact with charged particles. A single gamma ray photon created initially via a fusion process in the core, will 'live' for a very short time span before being annihilated. Another, or several, photons are then created in its place. That process continues, with net result of a very slow random-walk percolation of initial energy (not photon) from core to surface. What finally emerges and freely escapes to space is of course peaked in frequency around the optical band. Gravitational redshift is only a very tiny contributionm to that overall transformation of energy from very high energy photons to much lower energy ones at the solar surface. There is also an 'anomalous' ~ million degrees coronal emission which is extremely tenuous with negligible contribution to overall solar output.

Just to clarify on that popular but incorrect picture. In the QED interpretation, photons are continually being created and annihilated whenever they interact with charged particles. A single gamma ray photon created initially via a fusion process in the core, will 'live' for a very short time span before being annihilated. Another, or several, photons are then created in its place. That process continues, with net result of a very slow random-walk percolation of initial energy (not photon) from core to surface. What finally emerges and freely escapes to space is of course peaked in frequency around the optical band. Gravitational redshift is only a very tiny contributionm to that overall transformation of energy from very high energy photons to much lower energy ones at the solar surface. There is also an 'anomalous' ~ million degrees coronal emission which is extremely tenuous with negligible contribution to overall solar output.

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Do we observe any significant shift (red or blue) once it leaves the surface?